Daydream about cracking the RSA algorithm

Solemnly declare: This article is not the product of strict argumentation, but is just a reverie of "blind cat meets dead mouse". If this idea is feasible, it can actually be extended to more problems based on the decomposition of large prime numbers.

Before discussing, let's review the process of generating the public and private keys of the RSA algorithm:
1. Randomly select two large prime numbers p, q.
2. Calculate the public modulus n = pq of the public and private keys.
3. Calculate the Euler function φ(n) of modulus n.
4. Select a positive integer e such that 1 < e < φ(n) , and e and φ(n) are relatively prime.
5. Calculate d such that de ≡ 1 (mod φ(n) ).
6. n and e determine the public key, and n and d determine the private key.

The security of the RSA algorithm is based on the decomposition of large prime numbers. Conventional decomposition is indeed not easy, and I have no intention of violent decomposition. My so-called cracking idea is actually based on the following facts:
1. We can also have the computing power and storage capacity of computers owned by others.
2. Under the conditions of the previous point, we can also calculate the large prime numbers that others can calculate.
3. The product of prime numbers is unique.

After considering these three points, we can take the following approach:
1. Do our best to calculate all the prime numbers that our computer can find, assuming it is saved as prims = [2, 3, 5,,...] . Because under the condition of fact 1, we can think that the prime number selected by the other party must be in the prims table.
2. Calculate the product of each pair of prime numbers, and save it together with the corresponding prime factor (for the convenience of later query, it can also be stored according to the size of the product), assuming that it is saved as:
pm = [(4,(2,2)), (6, (2, 3)), (9, (3, 3)), (10, (2, 5)), ..., (15, (3, 5)), ...]

Because the prime number cannot change, we can perform these two steps when any computer is idle. At this time, our purpose is to simply calculate the prime number, how long it will run, etc. Don't think about it, Quan cultivates under the stage, raises troops for a thousand days, and waits for the use of troops for a while.
...
Finally, at some point, we obtained the other party's public key pair (n, e), and the pm table that we usually accumulated at this time came into use. By querying n in this table (the dichotomy method can be adopted), we can quickly know the corresponding prime factors p and q, and the remaining problems will naturally be solved!

It can be summed up in two sentences:
1. It takes a thousand days to keep an army, and a moment to use it.
2. One minute on stage, ten years off stage.